The study of integer representations as a sum powers is still very long standing problem. In this work, the representation cube introduced and investigated for non-zero distinct solution. Let \(a_1\), \(a_2\), \(a_3\), ... , \(a_n\) d be any positive integers such that - \(a_n\)-1= \(a_n\)-1 \(a_n\)-2 = \(a_2\) \(a_1\) d. This formulates some general results sums n cube. particular, research in...

In this paper we construct 16 free algebras of modular forms on type IV symmetric domains for some reflection groups related to the eight rescaled root lattices $$A_1(2)$$ , $$A_1(3)$$ $$A_1(4)$$ $$2A_1(2)$$ $$A_2(2)$$ $$A_2(3)$$ $$A_3(2)$$ $$D_4(2)$$ . As a corollary, prove modularity formal Fourier–Jacobi expansions these groups.

This paper deals with the boundary value problem involving the differential equation begin{equation*}     ell y:=-y''+qy=lambda y,  end{equation*}  subject to the standard boundary conditions along with the following discontinuity  conditions at a point $ain (0,pi)$  begin{equation*}     y(a+0)=a_1 y(a-0),quad y'(a+0)=a_1^{-1}y'(a-0)+a_2 y(a-0), end{equation*} where $q(x),  a_1 , a_2$ are  rea...

 Let $ a_0 (omega), a_1 (omega), a_2 (omega), dots, a_n (omega)$ be a sequence of independent random variables defined on a fixed probability space $(Omega, Pr, A)$. There are many known results for the expected number of real zeros of a polynomial $ a_0 (omega) psi_0(x)+ a_1 (omega)psi_1 (x)+, a_2 (omega)psi_2 (x)+...

Let G_1 and G_2 be simple connected graphs with disjoint vertex sets V(G_1) and V(G_2), respectively. For given vertices a_1in V(G_1) and a_2in V(G_2), a splice of G_1 and G_2 by vertices a_1 and a_2 is defined by identifying the vertices a_1 and a_2 in the union of G_1 and G_2. In this paper, we present exact formulas for computing some vertex-degree-based graph invariants of splice of graphs.

‎In this paper‎, ‎some algebraic and geometrical properties of the rank$-k$ numerical hulls of normal matrices are investigated‎. ‎A characterization of normal matrices whose rank$-1$ numerical hulls are equal to their numerical range is given‎. ‎Moreover‎, ‎using the extreme points of the numerical range‎, ‎the higher rank numerical hulls of matrices of the form $A_1 oplus i A_2$‎, ‎where $A_1...

Let $a_1,a_2,\dots,a_k$ be positive integers with $\gcd(a_1,a_2,\dots,a_k)=1$. ${\rm NR}={\rm NR}(a_1,a_2,\dots,a_k)$ denote the set of nonrepresentable in terms $a_1,a_2,\dots,a_k$. The largest integer $\max{\rm NR}$, number $\sum_{n\in{\rm NR}}1$ and sum NR}}n$ have been widely studied for a long time as related to famous Frobenius problem. In this paper by using Eulerian numbers, we give for...